1. The
power we use at home has a frequency of 60 Hz. The period of this sine wave can
be determined as follows:

__Solution__

T = 1/f = 1/60 = 0.0166 s

= 0.0166 x 10

^{3}ms = 16.6 ms
2. The
period of a signal is 100 ms. What is its frequency in kilohertz?

__Solution__

First we change 100 ms to seconds, and then
we calculate the frequency from the period (1 Hz = 10

^{−3}kHz).
100 ms = 100 x 10

^{-3}s = 10^{-1}s
f = 1/T = 1/10

^{-1}Hz = 10 Hz
= 10 x 10

^{-3}kHz = 10^{-2}kHz
Note:

*Frequency is the rate of change with respect to time.*

*Change in a short span of time means high frequency.*

*Change over a long span of time means low frequency.*

3. A
sine wave is offset 1/6 cycle with respect to time 0. What is its phase in
degrees and radians?

__Solution__

We know that 1 complete cycle is 360°. Therefore,
1/6 cycle is

1/6 x 360 = 60

^{o}
= 60 x 2π/360 rad = π/3 rad = 1.046 rad

Note:

*Phase describes the position of the waveform relative to time 0.*

*Three sine waves with the same amplitude and frequency, but different phases*

4. If
a periodic signal is decomposed into five sine waves with frequencies of 100,
300, 500, 700, and 900 Hz, what is its bandwidth? Draw the spectrum, assuming
all components have a maximum amplitude of 10 V.

__Solution__

Let f

_{h}be the highest frequency, f_{l}the lowest frequency, and B the bandwidth. Then
B = f

_{h}- f_{l}= 900 – 100 = 800 Hz
The spectrum has only five spikes, at 100,
300, 500, 700, and 900 Hz (see Figure).

Note:

*The bandwidth of a composite signal is the difference between the highest and the lowest frequencies contained in that signal.*

5. A
periodic signal has a bandwidth of 20 Hz. The highest frequency is 60 Hz. What
is the lowest frequency? Draw the spectrum if the signal contains all
frequencies of the same amplitude.

__Solution__

Let f

_{h}be the highest frequency, f_{l}the lowest frequency, and B the bandwidth. Then
B = f

_{h}- f_{l}=> 20 = 60 - f_{l}=> f_{l}= 60 – 20 = 40 Hz
The spectrum contains all integer
frequencies. We show this by a series of spikes (see Figure).

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